Equilibrium of Surfaces in a Vertical Force Field

In this paper we study $\varphi$-minimal surfaces in $\mathbb{R}^3$ when the function $\varphi$ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $\mathbb{R}^2$. We describe a full classification of complete flat embedded $\varphi$-minimal surfaces if $\varphi$ is strictly monotone and characterize $\varphi$-minimal bowls by its behavior at infinity when $\varphi$ has a quadratic growth.


Introduction
The equilibrium of a flexible, inextensible surface Σ in a force field F was given by Poisson [12, pp. 173-187] and when the intrinsic forces of the surface are assumed to be equal, the external force must have a potential P = e ϕ , that is, F = ∇P, for some smooth fuction ϕ on a domain of R 3 which contains Σ. In this case, the equilibrium condition is given in terms of the mean curvature vector H of Σ as follows: where ∇ is the gradient operator in R 3 and ⊥ denotes the projection to the normal bundle of Σ.
A surface satisfying (1.1) is called ϕ-minimal and it can be also viewed either as a critical point of the weighted volume functional where dA Σ is the volume element of Σ, or as a minimal surface in the conformally changed metric (1.3) G ϕ := e ϕ ·, · .
From this property of minimality, a tangency principle can be applied and any two different ϕ-minimal surfaces cannot "touch" each other at one interior or boundary point (see [3, Theorem 1 and Theorem 1a]).
In this paper, we are interested in the case that ϕ is invariant under a twoparameter group of translations in R 3 . Up to a motion in R 3 , we can assume that the external force field F is always a vertical field, that is, F ∧ e 3 = e ϕ ∇ϕ ∧ e 3 ≡ 0, with e 3 = (0, 0, 1) ∈ R 3 , where ϕ only depends on the third coordinate in R 3 and the mean curvature vector of Σ satisfies (1.4) H =φ e ⊥ 3 , here (˙) denotes derivate respect to the third coordinate.
Our objective in this paper is to develop a general theory of [ϕ, e 3 ]-minimal surfaces taking as a starting point some of the recent and important progress in theory of translating solitons and singular minimal surfaces in R 3 (see for instance [2,6,7,9,11,13,16]).
Nonetheless, the class of [ϕ, e 3 ]-minimal surfaces is indeed very large and much richer in whats refers to examples and geometric behaviors. Although new ideas are needed for its study, it will be necessary, in order to get classification results, to impose some additional conditions to the function ϕ. Here, as a general assumption we will consider ϕ strictly monotone, that is, ϕ :]a, b[⊆ R → R is a strictly increasing (or decreasing) function (1.6) and Σ ⊂ R 2 ×]a, b[. Despite these difficulties the results we present in this paper are given for ϕ in very general classes of regular functions.
The paper is organized as follows, in Section 2 we show some fundamental equations related to our family of surfaces and as a consequence we prove the non-existence of closed examples and two results about strictly convexity and mean convexity of [ϕ, e 3 ]-minimal surfaces.
Section 3 is devoted to the study and classification of embedded complete flat [ϕ, e 3 ]-minimal surfaces. We describe the so called [ϕ, e 3 ]-grim reapers and tilted [ϕ, e 3 ]-grim reapers and characterize them as the unique examples of embedded complete flat [ϕ, e 3 ]-minimal surfaces.
In Section 4 we study the existence and classification of rotational examples. We construct for ϕ in a very general class of functions (strictly increasing and convex) a family of [ϕ, e 3 ]-minimal bowls (which are strictly convex graphs) and [ϕ, e 3 ]-minimal catenoids with a wing-like shape (which resemble the usual translating catenoids in R 3 ).
Finally, Sections 5 and 6 are devoted to study [ϕ, e 3 ]-minimal surfaces when ϕ has a quadratic growth. We provide the asymptotic behavior of rotationally symmetric examples and characterize [ϕ, e 3 ]-bowls by their behavior at infinity.
Acknowledgements: The authors are grateful to Margarita Arias, José Antonio Gálvez and Francisco Martín for helpful comments during the preparation of this manuscript.

Some relevant equations
Here, we will give some local fundamental equations related to [ϕ, e 3 ]-minimal surfaces.
Let ψ : M −→ R 3 be a 2-dimensional [ϕ, e 3 ]-minimal immersion (maybe with a non empty boundary) with Gauss map N , induced metric g and second fundamental form A. We shall denote by ∇, ∆ and ∇ 2 , respectively, the Gradient, Laplacian and Hessian operators of g.
The mean curvature vector of ψ is defined by H = trace g A and the symmetric bilinear form A given by A(X, Y ) = − A(X, Y ), N , X, Y ∈ T Σ, is called scalar second fundamental form. The mean curvature function H will be the trace of A with respect to g. With this notation, (1.4) is equivalent to We will assume that ϕ satisfies (1.6) and let us introduce the height and angle functions, respectively, by: µ := ψ, e 3 , η := N, e 3 .
In the next result we show some relations involving H, µ and η: where A [2] and B are the symmetric 2-tensors given by the following expressions: for any vector fields X, Y ∈ T Σ and any orthonormal frame Proof. (1) Differentiating µ and η respect to any X ∈ T Σ, we get, ∇µ, X = dµ(X) = e 3 , X , ∇η, X = dη(X) = dN (X), e 3 = A(X, e 3 ).
But, by hypothesis, η is a nonpositive function, and so, from the strong maximum principle, if it vanishes anywhere then it vanishes everywhere, which concludes the proof. where So, can apply the maximum principle of Hamilton (see [14,Section 2]) and if there is an interior point of Σ where A has a null-eigenvalue then A must have a null-eigenvalue everywhere, which concludes the proof of the theorem.

Vertical graphs invariant by horizontal translations
Consider the [ϕ, e 3 ]-minimal vertical graph given by a function u which only depend on one variable, u = u(x), from (1.5) u must be a solution of the following ODE: In order to look for complete examples we will consider that is either a strictly increasing (or decreasing) function. Then, by taking z = ϕ(u) and u = tan(v), we obtain that (3.1) is equivalent to where h(z) =φ(ϕ −1 (z)). It is clear that e z cos(v) is constant along the solutions of (3.2) and from Figure 3.1, for each solution u of (3.1) there exists a unique x 0 ∈ R such that v(x 0 ) = 0 (it is not a restriction to assume that x 0 = 0).

Proof.
As the first item follows from (3.5).
On the other hand, by assuming thatφ is increasing and A similar discussion can be done whenφ is decreasing. (3.5) and Theorem 3.2, we can prove the following properties of the solutions, it is convex, symmetric about the y-axis and has a minimum at x = 0. Moreover, In particular, if Λ u0 < ∞, the graph of u is asymptotic to two vertical lines.
it is concave, symmetric about the y-axis and has a maximum at x = 0. Moreover, and, lim
Observe that, and it is the graph of the function If ϕ : R → R is a strictly increasing diffeomorphism, then Σ is either a vertical plane or a [ϕ, e 3 ]-grim reaper (maybe tilted) surface.
Proof. From basic differential geometry, Σ = α × Π ⊥ is a ruled surface and its Gauss map is constant along the rules, where α is a complete regular curve in a plane Π ⊂ R 3 .
Claim: Let L be a straight line of Σ and V L be the unit normal vector along L. If V L , e 3 = 0, then there exists a [ϕ, e 3 ]-grim reaper T L (tilted, if L is not horizontal) containing L and tangent to Σ along L.
Then, up to an appropriate rotation and dilatation, Σ is tangent to a [ϕ, e 3 ]grim reaper along a rule. The result follows from standard theory of uniqueness of solution for the ODE (3.1).
Proof of the claim. If L is horizontal then, after a rotation about the axis e 3 , we may assume that and there exists φ ∈] − π/2, π/2[ such that V L = (− sin φ, 0, cos φ). Then, as ϕ : R → R is a strictly increasing diffeomorphism, from (3.1), there exists a solution u L of (3.1)-(3.3) and The [ϕ, e 3 ]-grim reaper we are looking for is just a translation in the e 1 -axis of the grim reaper T u L associated to u L .
If L is not horizontal and p = L ∩ {z = 0}, then by rotation of center p and axis e 3 we may assume there exists θ ∈] − π/2, 0[ and α ∈ R, such that So, from (3.6) and (3.7), if we take the solution u L of (3.1)-(3.3) satisfying for some x 1 ∈ R, we conclude that our tilted [ϕ, e 3 ]-grim reaper is a translation in the e 1 -axis of the tilted [ϕ, e 3 ]-grim reaper obtained after rotation of angle θ around the e 2 -axis and dilation of 1/ cos θ the [ϕ, e 3 ]-grim reaper associated to u L .

[ϕ, e 3 ]-minimal surfaces of revolution
In this section and in a similar way to the case of translating solitons (see [8,9,11]), we are going to study the existence [ϕ, e 3 ]-minimal surfaces of bowltype and catenoid -type.

The singular case
In the rotationally symmetric case, the equation (1.5) reduces to the following ordinary differential equation for u = u(r), r = x 2 + y 2 : where ( ) denotes derivative respect to r and ϕ : Since (4.1) is degenerated, the existence and uniqueness of solution at r = 0 is not assured by standard theory. Multiplying by r we obtain that (4.1) also writes as, But, from [15, Theorem 2], a solution of (1.5) cannot possess isolated nonremovable singularities, hence, it is not a restriction to look for the existence of solutions of (4.2) with the following initial conditions: In this sense and by using a similar argument to [13, Proposition 2] we can assert for some R > 0 which depends continuously on the initial data and such that The following result allows us to compare rotational symmetric [ϕ, e 3 ]-minimal graphs, Hence, there exists > 0 such that d = u ϕ1 − u ϕ2 > 0 on ]0, [. If there exists r 1 > 0 satisfying d(r 1 ) ≤ 0, we can take r * := inf{r > 0 : d(r) < 0} so that d(r * ) = 0 and d (r * ) ≤ 0. But, from (4.1) and having in mind that which is a contradiction.

Bowls-type and catenoid-type examples
Now, we want to describe [ϕ, e 3 ]-minimal surfaces that are invariant under the one-parameter group of rotations that fix the e 3 direction. A such surface with generating curve the arc-lenght parametrized curve γ(s) = (x(s), 0, z(s)), s ∈ I ⊂ R is given by, The inner normal of ψ writes as where κ is the curvature of γ and by we denote derivative respect to s.
Along this section we will consider that ϕ : ]a, ∞[ −→ R is a strictly increasing and convex function, that is on ]a, ∞[.

Bowl-type examples
Here, we want to study the solutions of (4.10) with the following initial conditions, In this case G intersects orthogonally the rotation axis and we have the following result: Proof. First of all, we remark that the existence of γ around s = 0 is guaranteed from Proposition 4.1. Moreover, it is easy to see that x(s) = −x(−s), z(s) = z(−s) and θ(s) = −θ(−s) are also solutions of the same initial value problem (4.10)-(4.12). Hence, γ is symmetric respect to e 3 direction and we may consider only the case s ≥ 0.
Definition 4.6. If γ is a graph as in Theorem 4.5, we are going to say that the revolution surface with generating curve γ is a [ϕ, e 3 ]-minimal bowl.

Catenoid-type examples
Now, we want to study the solutions of (4.10) with the following initial conditions, From standard theory, the existence and uniqueness of solution to the problem (4.10)-(4.15) is guaranteed.   Proof. It is clear because θ < 0 on θ −1 ( π 2 ) and θ > 0 on θ −1 (0).    Figure 4.2 (right) with the annulus topology whose distance to axis of revolution is x 0 and whose generating curve γ is of winglike type see Figure 4.

Proof.
Ifφ has at most a linear growth, then there must be a constant c > 0 such thaṫ ϕ(u)/u ≤ c outside a compact set. Thus, from the inequality (4.14), when x is large enough the following inequalities hold, Integrating both members of the inequality (4.17), we get that, for some x 0 > 0.
Let's go to consider now that lim u→+∞φ (u) u α = M = 0 for some α > 1, and suppose that ω + = +∞. Then, from the Theorem 4.5 and Theorem 4.10, the real function f given by, has, for r large enough, a bounded and strictly monotone primitive F (u)(r). Hence, there exists a sequence {r n } +∞ such that  Proof of Claim 4.12. Assuming on the contrary, there exists δ > 0 and a sequence {s n } +∞ such that which together (4.19), says that f −1 (δ) is unbounded real subset containing a divergent sequence to +∞. But, from the equation (4.1), the function f satisfies the following differential equation and we obtain that there existsr ∈ f −1 (δ) such that f (r) > 1 for any r ∈ f −1 (δ), r ≥r, which is impossible because f −1 (δ) is unbounded.
From (4.20), Claim 4.12 and using that u diverges to +∞ we get that, for r sufficiently large, the following inequality holds , By integration of this expression, we conclude that ω + < +∞.
Remark 4.13. Notice that ω + = +∞ does not imply thatφ has at most a linear growth. For example, by takingφ(u) = u log(u) with u ≥ 1 and by the integration of both members in (4.14), we get that, for some x 0 > 0.

Asymptotic behavior of rotational examples
Clutterbuck, Schnürer and Schulze studied in [2] the asymptotic behavior of solitons rotationally symmetric. They proved that the problem has a unique C ∞ -solution u on [R, ∞[. Moreover, as r → ∞, u has the following asymptotic expansion Due to the arbitrariness of the problem (4.1) it is impossible to find a general asymptotic behavior of their solutions because if you consider any strictly convex smooth function u = u(r), r > R, one can find a function ϕ such that u is a solution of (4.1).
Proposition 4.11 motivates to consider ϕ :]a, +∞[−→ R a regular function satisfying (4.11) and with a quadratic growth, that is, with the following asymptotic behavior, In this case, we are going to generalize the result in [2] to the following problem, It is also clear that v From Remark 5.1, in order to study the asymptotic behavior of u φ(u) , it is not a restriction to assume that β > 0.
Proof of Claim 5.4. Assuming on the contrary, if u (r) − ζ R (r) ≤ 0 for any r > R, then the following inequalities holds, Integrating, we can find a finite radius r such that u → +∞ as r → r, getting a contraction since the solution u is defined for all r > r 0 . On the other hand, as d(r) > 0 for any r ∈]R, s[ we have by integration of d that, and (4.11) gives that d (s) >φ(u(s)) −φ(ζ R (s)) > 0 which is a contradiction. Thus, d(r) > 0 for r large enough and by using the inequality (5.6), we get, Moreover, from the previous formula (5.12) and L'Hôpital's rule, we also get that, lim r→+∞ log φ 2 (u(r)) αr 2 = 1 andφ(u) has the following asymptotic expansion, (5.13)φ(u)(r) = e 1 2 α r 2 +o(r 2 ) .
Now, choosing R large enough, the equation (5.14) and the inequalities (5.7) and (5.15) give, Using the conditions (4.11) and the asymptotic behavior (5.13), R may be chosen large enough so thaṫ Thus, if R is large enough and r ≥ R where V 1 (r) ≤ −ε, then V 1 (r) ≥ c > 0. Hence, V 1 (r) ≥ −ε for r large enough and we conclude the proof.
In a similar way we may prove that λ(r) ≤ −α−ε for r sufficiently large. Now (5.5) follows from (5.12), (5.13) and Claims 5.5 and 5.6. we have the following asymptotic expansion: Proof. Arguing as in Theorem 4.5, Theorem 4.10 and Proposition 4.11 , (5.3) has a unique C ∞ -solution u on [r 0 , ∞[ which is strictly convex function satisfying that lim r→∞ u(r) = ∞. Moreover, as Claims 5.3 and 5.4 also work in this case, we have the following asymptotic expansion where V 1 verifies the same differential equation (5.14), is also nonpositive and V 1 (r) → 0. Moreover, from (5.2),φ writes as Consider now the new function V 2 (r) = rφ 2 (u)(r)V 1 (r). Then Corollary 5.8. Assume thatφ has the following expansioṅ ϕ(u) = αu + β + ∞ n=1 a n u n , a n ∈ R, (5.23) where either α > 0 and the first non-vanishing a k is positive or α = 0, β > 0 and the first non-vanishing a k is negative. Then, for any solution u of the problem (5.1) we have the following asymptotic behavior, if α > 0 and, up to a constant where G is the strictly increasing function given by Proof. If α > 0, from (5.12) and (5.13) we can write, where Υ = (φ − α)r +φV 1 . Hence, as the first non-vanishing a k is positive, for r large enough Υ is a decreasing function in r such that −∞ < c = lim r→+∞ Υ(r) otherwise from Claim 5.6, (5.23), (5.26) and by using L'Hôpital's rule, we have that, which is a contradiction.
Applying again L'Hôpital's rule to lim r→+∞ e 2Υ − e 2c e −αr 2 , we havė If α = 0 then, the condition (5.18) follows from (5.20) and we have that Now, by taking V 3 (r) = (V 2 (r) + 1)r 2 we get But, from (5.20) and L'Hôpital's rule, we obtain thus, by working as in Claim 5.6 we prove that and (5.25) follows from integration in the above expression.
6 Uniqueness of bowl-type's solutions Along this section ϕ :]a, +∞[−→ R will be a regular function satisfying the expansion (5.23).
For any θ ∈ [0, 2π[ we consider v = (cos θ, sin θ, 0) and denote by Π v (t) the vertical plane Definition 6.1. Let M 1 and M 2 be two arbitrary subsets of R 3 . We say that M 1 is on the right hand side of M 2 respect to Π v (t) and write M 1 ≥ v M 2 if and only if for every point q ∈ Π v (t) such that, we have the following inequality, where π : R 3 → Π v (t) denotes the orthogonal projection on Π v (t).
For an arbitrary subset M of R 3 we also consider the following subsets: From Corollary 5.8 it is natural to study [ϕ, e 3 ]-minimal surfaces whose behavior at infinity is of rotational type. To be more precise, Definition 6.2. We say that a [ϕ, e 3 ]-minimal end Σ is smoothly asymptotic to a rotational-type example if Σ can be expressed outside a ball as a vertical graph of a function u Σ so that, according to α is either positive or zero, one of the following expressions holds where C is a positive constant or up to a constant, if α = 0 and β > 0.
Let Σ be an embedded [ϕ, e 3 ]-minimal surface Σ with a single end smoothly asymptotic to a bowl-type example. Then, there exists R > 0 large enough such that Σ∩(R 3 \B(0, R)) is the vertical graph of a function u Σ verifying either (6.2) if α > 0 or (6.3) if α = 0 and β > 0. Lemma 6.3. There exists r 1 > R such that if t > r 1 then Σ + (t) is a graph over Π v (t).
Hence, there exists r 1 large enough such that if x, v ≥ r 1 , then (du Σ ) x ( v) > 0 and, in this case, the Lemma follows because Σ is embedded and Σ + (r 1 ) ∪ π(Σ + (r 1 )) bounds a domain in R 3 . When α = 0 a similar argument with (6.3) also works.
When α = 0, we can estimate G(u * t )(x) − G(u Σ )(x) as in [9, Claim 1, Step 3] and to use that G is a strictly increasing function.
Theorem 6.5. Let Σ be a complete properly embedded [ϕ, e 3 ]-minimal surface in R 3 with a single end that is smoothly asymptotic to a bowl-type example. Then the surface Σ is a [ϕ, e 3 ]-minimal bowl.
Proof. The main idea is to use the Alexandrov's reflection principle, [1], for proving that Σ is symmetrical with respect to Π v (0). For proving that, it is not difficult to see that Lemma 6.3 and Lemma 6.4 are the fundamental facts we need to check that all the steps in the proof of Theorem A in [9] can be adapted to our case and for getting to prove that 0 ∈ A were A := {t ≥ 0 : Σ + (t)is a graph over Π v (t) and Σ * A symmetrical argument gives that Σ * − (0) ≤ v Σ + (0). Hence, Σ * + (0) = Σ − (0) and Σ is symmetric respect to the plane Π v (0). As v = (cos θ, sin θ, 0) represents any unit horizontal vector, Σ would be a revolution surface touching the axis of revolution, that is, a [ϕ, e 3 ]-minimal bowl.

Concluding remarks
(i) It would be interesting to give a clasification of [ϕ, e 3 ]-maximal surfaces in the Lorentz-Minkowski space L 3 using the Calabi's Type correspondence of [10].
(ii) From the minimality of these surfaces in the conformally changed metric G ϕ it is reasonable to think whether classical theorems on minimum surfaces are true. For example: if we consider an one-parametric family of winglikes {W} R and taking the size of the neck R converging to zero, then they converging to a double recovering of a punctured bowl. Thus, it is posible that a result as the half-space theorem holds for [ϕ, e 3 ]-minimal surfaces.
(iii) Other related problem with the theory of minimal surfaces, that we could be study is the Jenkins-Serrin problem for [ϕ, e 3 ]-minimal graphs when the metric e ϕ ·, · is complete. For this purpose [4,5] are interesting references.